Experiments on Gaussian beams and vortices in optically induced photonic...

PROOF COPY [57713] 014507JOB

PROO

F COPY [57713] 014507JO

B Experiments on Gaussian beams and vortices in

optically induced photonic lattices

Zhigang Chen, Hector Martin, and Anna Bezryadina

Department of Physics and Astronomy, San Francisco State University, San Francisco, California 94132

Dragomir Neshev and Yuri S. Kivshar

Nonlinear Physics Centre and Centre for Ultrahigh-Bandwidth Devices form Optical Systems, Research School ofPhysical Sciences and Engineering, Australian National University, Canberra 0200, Australia

Demetrios N. Christodoulides

School of Optics/CREOL, University of Central Florida, Orlando, Florida 32816

Received October 28, 2004; accepted January 3, 2005

We investigate experimentally the propagation of fundamental Gaussian beams and vortices in a two-dimensional photonic lattice optically induced with partially coherent light. We focus on soliton–lattice inter-actions and vortex–lattice interactions when the lattice is operated in a nonlinear regime. In this case a host ofnovel phenomena is demonstrated including soliton-induced lattice dislocation–deformation, soliton hoppingand slow-down, and creation of structures akin to optical polarons. In addition, we observe that the nonlinearinteraction between a vortex beam and a solitonic lattice leads to lattice twisting due to a transfer of the an-gular momentum carried by the vortex beam to the lattice. Results demonstrating a clear transition from dis-crete diffraction to the formation of two-dimensional, discrete fundamental and vortex solitons in a linear lat-tice are also included. © 2005 Optical Society of America

OCIS codes: 190.5940, 270.5530, 190.4420.

1. INTRODUCTIONRecently there has blossomed an interest in the study

of light propagation in periodic structures including in-trinsic localized modes and discretizing light behavior inlinear and nonlinear waveguide lattices.1,2 Closely spacedwaveguide arrays represent a typical example of opticalperiodic structures that have a strong link with theemerging science and technology of nonlinear photoniccrystals. In such array structures the collective behaviorof wave propagation exhibits many fascinating phenom-ena that are also encountered in other discrete nonlinearsystems. For example, optical structures with a periodicchange in the material refractive index can profoundly af-fect the diffraction properties of light. Even in a linearwaveguide array, due to evanescent coupling betweennearby waveguide sites, a light beam experiences discretediffraction. If the waveguide array is embedded in a non-linear medium, more interesting wave behavior is ex-pected to occur. A typical example is the balance betweendiscrete diffraction and nonlinear self-focusing that leadsto optical self-trapped states better known as discretesolitons.3 In nonlinear optics, discrete solitons were firstdemonstrated in one-dimensional (1D) AlGaAs semicon-ductor waveguide arrays in 1998,4,5 and they have sinceattracted considerable research interest. In part, this isbecause such discrete solitons are expected to exist also ina variety of other discrete nonlinear systems, includingthose in biology, solid-state physics, and Bose–Einsteincondensates.6,7 Apart from fundamental interest, these

discrete solitons have been also proposed as good candi-dates for all-optical blocking and switching of signals intwo-dimensional (2D) waveguide networks.8

Yet in spite of the exciting prospects for applications,the design and manufacture of such 2D waveguide arrays(with bandgaps in k space) is a difficult, expensive, andtime-consuming task. Additionally the materials used forimplementing such periodic systems normally do not pos-sess strong properties for nonlinear self-action of light.Therefore a new approach is needed to realize stronglynonlinear periodic structures in higher dimensions. Re-cently a theoretical study suggested that it was possibleto create reconfigurable periodic structures by use of op-tical induction in photorefractive crystals with strong an-isotropic nonlinearity.9 This soon led to the experimentalobservation of 1D and 2D discrete solitons in such opti-cally induced waveguide lattices, as first reported in Refs.10,11 and then in Refs. 12,13. Unlike AlGaAs waveguidearrays, the optically induced lattices can be readily madein two or even three dimensions. This allows the study oflattice solitons in higher dimensions.

Meanwhile photonic lattices optically induced by pixel-like spatial solitons have been created with amplitudemodulation of partially spatially incoherent light14,15 aswell as with phase engineering of coherent light,16,17 lead-ing to study of optical control and manipulation of indi-vidual channels in a large array of solitonic lattices. Byexploiting the anisotropic photorefractive nonlinearity,the lattices could be conveniently operated in either the

Chen et al. Vol. 22, No. 7 /July 2005 /J. Opt. Soc. Am. B 1

APC:#1

0740-3224/05/0700001-03/$15.00 © 2005 Optical Society of America



PROO

F COPY [57713] 014507JO

B linear or the nonlinear regime. If the probe beam and thelattice beam are extraordinarily polarized (e polarized),they both experience high nonlinearity as they propagatetogether and thus interact strongly. In this configurationthe lattice beam itself tends to form arrays of spatial soli-tons, so the soliton-induced waveguide lattice is said to bein the “nonlinear” regime, which becomes considerablymore susceptible to soliton-induced deformation.15 This inturn brings about the interesting possibility of studyingoptical soliton-lattice interactions that might exhibitmany of the basic characteristic features of other physicalprocesses such as those encountered in polaronexcitation–formation in solid-state physics. In particular,soliton-induced lattice dislocation–deformation,13 latticecompression waves,18 and a novel type of composite band-gap solitons19 are expected to occur in such nonlinear lat-tices.

On the other hand if the lattice beam is ordinarily po-larized (o polarized) while the probe beam is e polarized,the probe would experience strong nonlinearity but theinduced lattice would remain nearly invariant duringpropagation due to the anisotropic property of the nonlin-ear crystal. In this case the waveguide lattice is said to bein the “linear” regime. This latter configuration can beused as a test bench for demonstration of many nonlinearwave propagation phenomena in periodic systems, includ-ing discrete solitons. When the lattice beam is createdwith amplitude modulation and partial coherence13

rather than by coherent multibeam interference,10–12 anenhanced stability of the lattice structure can be achievedin both the linear and nonlinear regimes as a result ofsuppression of incoherent modulation instabilities, aswell as reduced sensitivity to ambient fluctuations. In factit is in such optically-induced, partially coherent photoniclattices that a clear transition from 2D discrete diffractionto formation of 2D discrete solitons has been successfullydemonstrated.13

This convenient way of creating reconfigurable photo-nic structures and of successfully demonstrating discreteself-trapping of light has opened the door for the observa-tion of many distinctive phenomena in 2D periodic mediathat have no counterpart in the bulk. These include, forexample, discrete vortex solitons,20,21 discrete solitontrains,22 and discrete dipole and vector solitons in 2Dlattices.23,24 In addition, other interesting phenomenathat have been predicted or observed in 1D waveguide lat-tices, such as discrete random-phase solitons, discretemodulation instability, grating-mediated waveguiding,discrete soliton interaction, and generation and control ofspatial gap solitons,25–30 are expected to occur in 2D lat-tices as well, with even much richer dynamics yet to beexplored.

In this paper we present our experimental work on thestudy of 2D fundamental Gaussian beams and vorticespropagating in optically induced linear and nonlinearphotonic lattices induced by partially coherent light. Weshall focus mainly on the nonlinear interaction between aGaussian–vortex beam and a 2D solitonic lattice. Whenthe lattice is operated in the nonlinear regime, the simul-taneous propagation and interaction between theGaussian–vortex beam and the lattice leads to the obser-vation of new phenomena. This includes, for example,

soliton-induced lattice dislocation–deformation, vortex-induced lattice twisting, and the creation of optical struc-tures that are somewhat analogous to polaron states insolid-state physics. The Gaussian beam itself tends toform a soliton in the self-focusing nonlinear medium, butits propagation is strongly affected by the interactionwith the nonlinear lattice, leading to its transverse veloc-ity slowdown in the lattice. The vortex beam with unit to-pological charge tends to break up into four filamentswhen launched appropriately into the nonlinear lattice,but the filaments are somewhat confined by the lattice po-tential so they cannot rotate freely. Vortices with higher-order charges are less confined by the nonlinear lattice,and instead they often cause stronger deformation of thelattice. For completeness, we will also include our recentexperimental results for Gaussian–vortex beams propa-gating as probes in linear waveguide lattices.

Conversely when the lattice is operated in the linear re-gime so that it does not itself experience strong nonlinear-ity during propagation, we observe that the probe beamevolves from discrete diffraction to 2D fundamental andvortex discrete solitons as the level of nonlinearity for theprobe is increased. Our experimental results for discretesolitons are in good agreement with the theoretical analy-sis of these effects.

2. EXPERIMENTAL SETUP AND CREATIONOF 2D LATTICESThe experimental setup for our study is illustrated in Fig.1. A partially spatially incoherent light beam ��=488 nm� is generated by converting an argon-ion laserbeam into a quasi-monochromatic light source with a ro-tating diffuser. Such a diffused laser source has the ad-vantage of providing an incoherent beam with a control-lable degree of spatial coherence and intensity suitablefor experiments with partially coherent light.31,32 A biasedphotorefractive crystal (SBN:60, 5�5�8 mm3, r33=280 pm/V and r13=24 pm/V) is employed to provide asaturable, self-focusing noninstantaneous nonlinearity. Togenerate a 2D waveguide lattice we use an amplitudemask to spatially modulate the otherwise uniform inco-herent beam after the diffuser. The mask is then imagedonto the input face of the crystal, thus creating a pixellikeinput intensity pattern.15 This lattice beam can be turnedinto either extraordinarily or ordinarily polarized lightwith a half-wave plate as necessary. A Gaussian beamsplit from the same laser but without passing through the

Fig. 1. Experimental setup. PBS, polarizing beam splitter;SBN, strontium barium niobate. The vortex mask is used only forexperiments with a vortex beam.

2 J. Opt. Soc. Am. B/Vol. 22, No. 7 /July 2005 Chen et al.



PROO

F COPY [57713] 014507JO

B

diffuser is used as the probe beam propagating along withthe lattice. For the vortex experiment the probe beam issent through a transmission vortex mask (bottom path)and then focused onto the crystal input face, propagatingcollinearly with the lattice. In addition a uniform ordi-narily polarized background beam (top path) as createdwith another rotating diffuser is used as incoherent back-ground illumination for fine-tuning the nonlinearity. Theinput and output faces of the crystal are imaged onto aCCD camera. The beam separation in a coupled or inter-acting state is achieved by blocking one of the componentsand quickly recording the other one, taking advantage ofthe noninstantaneous response of the crystal.

In our experiment the probe beam is e polarized and“fully” coherent, while the lattice beam is partially coher-ent with either extraordinary or ordinary polarization asneeded. In general, in an anisotropic photorefractive crys-tal, the nonlinear index change experienced by an opticalbeam depends on its polarization as well as its intensity.Under appreciable bias conditions, i.e., when the photore-fractive screening nonlinearity is dominant,9 this indexchange is given approximately by �ne= �ne

3r33E0 /2��1+I�−1 and �no= �no

3r13E0 /2��1+I�−1 for e polarized and opolarized beams, respectively. Here E0 is the applied elec-tric field along the crystalline c axis (x direction), and I isthe intensity of the beam normalized to the backgroundillumination. As a result of the difference between thenonlinear electro-optic coefficients r33 and r13 in theSBN:60 crystal we used, �ne is more than 10 times largerthan �no under the same experimental conditions. Thuswhen it is e polarized, the lattice beam experiences a non-linear index change comparable to that of the probe beam,whereas it evolves almost linearly when it is o polarized.In this way by exploiting the anisotropic photorefractivenonlinearity, the lattices can be conveniently operated ineither the linear or nonlinear regime.

Typical examples of 2D square lattices created by opti-cal induction in our crystal with both o polarized and e po-larized beams are shown in Fig. 2. Our experiment showsthat the linear square lattices generated with an o polar-ized partially coherent beam are stable and robust, evenat a small lattice spacing of 20 �m or less. However, tocreate a nonlinear (solitonic) square lattice with the e po-larized beam, it is a challenge to obtain a stable latticewithout distortion at such small lattice spacing. In addi-

tion because of the anisotropic self-focusing nonlinearity,the pixels (or waveguides) tend to merge in the y directionif the square lattice is oriented in horizontal–vertical di-rections. The diagonal orientation of the square lattice(i.e., its principal axes oriented in the 45° directions rela-tive to the x and y axes) favors stable lattice formation asa result of the enlarged y separation between pixels, es-pecially when the lattice spacing is small. Figure 2(a)shows a stable 2D linear lattice with 20 �m spacing cre-ated with an o polarized beam, whereas Fig. 2(b) showsthe corresponding pattern when the same lattice beamwas changed to e polarized with all other conditions un-changed. Clearly, the lattice is strongly distorted at thisspacing in the nonlinear regime. The reason for this isquite intuitive. The e polarized lattice experiences astrong nonlinear self-focusing effect, thus each lattice sitetends to form a pixel-like soliton itself. When the latticespacing is small, the size of each soliton pixel is alsosmall. With the photorefractive screening nonlinearity,the formation of solitons must satisfy certain conditionsbetter described by the soliton-existence curve.33 In es-sence, forming a smaller soliton requires a stronger non-linearity, which is usually achieved by increasing the biasfield across the nonlinear crystal. However, with a stron-ger nonlinearity, an e polarized lattice beam tends to suf-fer from stronger modulation instability as driven bynoise such as defects and striations in the crystal, whichafflicts the formation of a stable solitonic lattice. This iswhy we introduce partial coherence in the lattice beam.As has been previously predicted34 and experimentallydemonstrated,14,35,36 partial coherence can result in sup-pression of modulation instabilities. Therefore, to create anonlinear lattice with enhanced stability often requiresfine-tuning of such experimental parameters as the biasfield, lattice spacing, lattice beam intensity, and spatialcoherence. If the spacing of a nonlinear lattice is too small(20 �m or less), it is difficult to obtain a stable, denselypacked solitonic lattice in our photorefractive crystal evenwith a partially coherent beam. In fact the lattice beamwould go haywire [as shown in Fig. 2(b)] because of strongmodulation instability and interaction between neighbor-ing lattice sites. At larger lattice spacings, nonlinear lat-tices of pixel-like spatial solitons can indeed be estab-lished by fine-tuning the nonlinearity. Figure 2(c) shows atypical example of a stable nonlinear lattice (at 37 �m

Fig. 2. Output intensity patterns of a 2D square lattice as created by optical induction. (a) Stable o-polarized linear lattice at 20 �mspacing, (b) its corresponding output when the lattice beam is turned to e polarized. (c) Stable e-polarized nonlinear lattice at 37 �mspacing, (d) its corresponding output when the nonlinearity is too high.


APC:#2



PROO

F COPY [57713] 014507JO

B spacing) obtained in our experiment, and Fig 2(d) showsthe corresponding lattice when the background illumina-tion and–or the partial coherence of the beam is removedso that the nonlinearity becomes too strong

In the following the square lattice will be oriented di-agonally for the experiments on discrete solitons andvortex–lattice interaction, while it will remain in ahorizontal–vertical orientation for the experiments onsoliton-lattice interaction.

3. GAUSSIAN BEAMS IN LINEAR ANDNONLINEAR PHOTONIC LATTICESA. Formation of Discrete Solitons in Linear LatticesFirst we present our experimental results of a 2D Gauss-ian beam used as a probe in a 2D linear waveguide lat-tice. In this case the lattice beam is o polarized while theGaussian beam is e polarized. As discussed earlier, thelattice in this configuration “sees” only a weak nonlinear-ity as compared with that experienced by the e polarizedprobe beam and therefore remains nearly invariant as thebias field increases. At the same time, the partially coher-ent lattice provides linear waveguiding for the probebeam. Typical experimental results are presented in Fig.3, for which a lattice with a small spacing of 20 �m [asshown in Fig. 2(a)] is first generated. The probe beam(whose intensity is 4 times weaker than that of the lat-tice) is launched into one of the lattice sites, propagatingcollinearly with the lattice. As a result of weak couplingbetween closely spaced waveguides, the probe beam un-dergoes discrete diffraction when the nonlinearity is low[Fig. 3(c)], which clearly shows that most of the energyflows from the center toward the diagonal directions ofthe lattice. However, the probe beam forms a 2D discretesoliton at an appropriate (high) level of nonlinearity [Fig.3(d)], so that most of its energy is concentrated in the cen-ter and the four neighboring sites along the principal axesof the lattice. These experimental results are in goodagreement with expected behavior from the theory of 2Ddiscrete systems.12,13 We note that in this case the lattice

itself behaves just like a fabricated linear waveguide ar-ray, and it is not considerably affected by the weak probeor the increased bias field.

B. Interaction of a Soliton with a Nonlinear LatticeNext we present the results of interaction of a Gaussianbeam (as a control–probe beam) with a nonlinear latticeinduced by 2D pixel-like spatial solitons. In this case boththe Gaussian beam and the lattice beam are e polarized.Under appropriate conditions a stable soliton lattice is es-tablished, and then the control–probe beam that tends toform a soliton itself is launched into the lattice. If the in-tensity of the Gaussian beam is much weaker than that ofthe lattice, the Gaussian beam behaves more as a probeand is simply guided by the soliton channel into which itis targeted without much influence on the propagation ofthe solitonic lattice. However, once the intensity of theGaussian beam becomes comparable with that of the lat-tice, it behaves more like a control beam and thus plays arole in the interaction process. For such a configuration,optical control and manipulation of individual solitonchannels by off-site interaction at relatively large latticespacing (70 �m or more) have been established previouslyeither by use of a partially coherent lattice15 or by phaseengineering of a coherent lattice.16,17 Here we focus onnovel aspects of behavior concerning soliton-lattice inter-actions when the input soliton is aimed at one of the soli-ton pixels (on-site interaction) and when the lattice spac-ing is relatively small (less than 70 �m). Of specialinterest is the interaction resulting from a probe–controlsoliton that carries an initial transverse momentum.

Typically when the Gaussian beam is launched at oneof the lattice sites with a small tilt angle relative to thepropagation direction of the lattice, we observe lattice dis-location due to soliton dragging. Figure 4 shows an ex-ample of such an interaction. For this experiment the co-herence length of the partially coherent lattice is 40 �m,and its intensity is about 3 times higher than that of thebackground illumination. The lattice spacing is 65 �m,and the intensity FWHM of each pixel is 17 �m. The 2Dsolitonic nonlinear lattice was created at a bias field of

Fig. 3. Gaussian beam propagating in optically-induced linear lattices, showing formation of a 2D discrete soliton. Top, 3D intensityplots, bottom, 2D transverse patterns of (a) the probe beam at crystal input, (b) output of normal diffraction without the lattice, (c) outputof discrete diffraction at low nonlinearity (bias field, 900 V/cm), (d) output of discrete soliton at high nonlinearity (bias field, 3000 V/cm).




PROO

F COPY [57713] 014507JO

B

2400 V/cm. At this bias field, the Gaussian beam with anintensity approximately equal to that of the lattice formeda soliton itself. When this control soliton was aimed intoone of the lattice sites at a shallow angle in four differentdirections, the lattice overall remained uniform except fora dislocation created by the soliton (Fig. 4, top panels). Inall cases the nonlinear lattice and the Gaussian beamhave interacted for a long period of time and the crystalhas reached a steady state. By capitalizing on the nonin-stantaneous response of the crystal, the control beam andthe lattice were monitored separately.

The photographs in the top panel of Fig. 4 were takenimmediately after the Gaussian beam was blocked in aninteracting state. The crosshair marker serves as a pointof reference to indicate how much the “target” pixel hasshifted from its original location at the output of the crys-tal as a result of soliton dragging. This soliton-induceddislocation in the lattice disappeared after the Gaussianbeam was blocked for more than 5 min and the crystalhad reached a new steady state; so the lattice was re-stored in the absence of the control beam. Clearly, thecontrol beam had the tendency to drag the interactingsoliton pixel to any direction at will. Meanwhile the con-trol beam itself retained its soliton identity, but its lateralshift (at the crystal output) was reduced significantly be-cause of the interaction with the lattice, indicating a slow-down in its transverse velocity (Fig. 4, bottom panels). Inthe bottom panel of Figs. 4(a) and 4(b), the control beamtraveled 68 �m in the horizontal (left and right) direc-tions without the lattice (the spot far away from the cen-ter denoted B), but it traveled only about 26 �m when in-teracting with the lattice (the spot close to the centerdenoted A). These two photographs were taken separately(in two separate experiments with and without latticewhile keeping all other conditions unchanged) and thensuperimposed in the same figure. Slowdown of the solitontransverse velocity as caused by the lattice is evident. In-terestingly enough, when the control beam was aimed in

vertical (up and down) directions at a 0.8° launchingangle, the slowdown of its transverse velocity was lesssignificant. However, in this case, soliton hopping mighthave occurred since the probe soliton jumped from the“target” lattice site to the next closest lattice site. In thebottom panels of Figs. 4(c) and 4(d) it can be seen that thecontrol soliton (denoted A) ended up at the output locationone site above the “target” lattice site. By comparing withFigs. 4(a) and 4(b) where no hopping has occurred at a0.5° launching angle, we expect that this kind of solitonhopping might be more evident at larger launching anglesas the control–probe soliton jostles its way through thenonlinear lattice.

The above-mentioned soliton-induced lattice disloca-tion became even more pronounced when the lattice spac-ing was decreased further and–or the probe intensity wasincreased. In addition other interesting phenomena wereobserved as the initial conditions for soliton-lattice inter-action were varied, including generation of lattice defects,lattice deformation, and lattice compression. Figure 5shows an example of how the intensity of the probe soli-ton plays its role in the dynamics of interaction. For theseexperiments the spacing of the lattice was kept at�50 �m, and the intensity of the probe was increasedgradually with all other experimental conditions un-changed. The results in Fig. 5 were obtained when theprobe beam was aimed into one of the lattice sites butwith a tilt angle of 0.8° toward the right (+x direction). Inthe top panels of Fig. 5 it can be seen clearly that the lat-tice dislocation caused by the control soliton becamestronger as the intensity of the soliton was increased. Infact, besides the pixel initially targeted by the control soli-ton (the central one as referenced by the cross-hair), otherpixels in close proximity (the right one and bottom one)were also dislocated from their initial positions. Pixelsfarther away in the direction of the initial transverse mo-mentum of the control soliton have also been affected [seeFig. 5(c)] as if the lattice were compressed slightly by the

Fig. 4. Top, soliton-induced lattice dislocation when a probe soliton was launched into a nonlinear lattice toward the (a) right, (b) left,(c) up, (d) down directions. The launching angle relative to the direction of lattice propagation is 0.5° for (a), (b) and 0.8° for (c), (d).Bottom, superimposed images of the probe beam taken with (denoted by “A”) and without (denoted by “B”) the lattice. (a), (b) show aslowdown in the transverse velocity of the probe soliton, (c), (d) show soliton hopping from site to site under the influence of the lattice.The lattice spacing is 65 �m in all cases.




PROO

F COPY [57713] 014507JO

B control soliton. This kind of lattice compression (orexcitation-lattice compression waves) has been observedalso during a process of interaction between a stripe soli-ton and a 2D nonlinear lattice.18 For comparison the lat-tice restored to steady state after removing the controlbeam is shown in Fig. 5(d). Clearly, the increased latticedistortion shown in Figs. 5(a)–5(c) results from the inter-action with the control soliton. In the bottom panels ofFig. 5, the location of the control soliton at crystal outputis shown. When the lattice was absent [bottom panels ofFigs. 5(a)–5(c)] the control soliton experienced an increas-ing self-bending toward the crystalline c axis (+x direc-tion) as its intensity was gradually increased. However,when interacting with the nonlinear lattice, suchdiffusion-induced self-bending effect37 was strongly sup-pressed, and the soliton propagated through the latticewith a much lower transverse velocity, as shown in thebottom panel of Fig. 5(d).

For comparison the experiments of Fig. 5 were re-peated under the same conditions except that the controlbeam was launched with a tilt angle of 0.8° toward theleft (−x direction). The corresponding results are pre-sented in Fig. 6 where, from (a) to (c), the top panel showsthe lattice dislocation–distortion as the intensity of thecontrol soliton was increased gradually, and the bottompanel shows the corresponding free propagation (no lat-tice present) of the control soliton. Clearly, the centralpixel that was targeted moved away, and other pixelsnearby (the left one and bottom one) were also dislocatedas compared with the unperturbed solitonic lattice [Fig. 6,top]. When the intensity of the control soliton was strong,pixel merging and pixel annihilation was achieved. A dif-ference was observed between launching the control beamwith and against the crystalline c axis that may be attrib-uted to the anisotropic soliton self-bending effect. We notethat the pixel annihilation was reproduced in several

Fig. 5. Top, soliton-induced lattice dislocation when the intensity of a probe beam was increased gradually. The launching angle relativeto the direction of lattice propagation is �0.8° toward the right. The intensity of a probe beam normalized to that of the lattice beam is(a) 0.9, (b) 1.0, (c) 1.1; (d) shows the restored lattice after the probe beam is turned off. Bottom, the corresponding intensity patterns ofthe probe soliton when the lattice beam is turned (a)–(c) off, (d) on. The lattice spacing is 50 �m in all cases.

Fig. 6. Same as in Fig. 5, except that the launching angle relative to the direction of lattice propagation is �0.8° toward the left, andthe intensity ratio between the probe soliton and the lattice is (a) 0.9, (b) 1.1, (c) 1.3.




PROO

F COPY [57713] 014507JO

B

other experiments when the control soliton had an initialtransverse momentum in the −x direction. While the out-put location of the control soliton was strongly dependenton its intensity in the absence of the lattice [Figs.6(a)–6(c), bottom], it stayed nearly the same when thecontrol soliton propagated through the nonlinear latticewith three different intensities [Fig. 6, bottom]. As a re-sult of interaction and energy exchange with more thanone soliton pixel in the lattice, the shape of the controlsoliton was also slightly distorted.

We have also investigated the case when the controlsoliton was launched into one of the lattice sites with noinitial transverse momentum, i.e., the soliton beam andthe lattice beam propagate along the same direction. Inthis case, strong lattice deformation, including generationof polaronlike structures, was also observed due to non-linear soliton–lattice interaction. Typical examples areshown in Fig. 7, where from (a) to (d) the intensity of theprobe soliton was increased gradually while keeping allother conditions unchanged. The lattice spacing is�55 �m for the top panels and �40 �m for the bottompanels. As the intensity of the probe soliton was increasedto a point comparable with that of the nonlinear lattice, apolaronlike induced structure was observed in which theprobe soliton dragged toward it some of the neighboringsites while pushing away the other [Fig. 7(c)]. When theprobe beam intensity was increased to be much higherthan that of the lattice, the lattice structure becamestrongly deformed in such a way that the site dislocationsextended beyond the immediate neighborhood [Fig. 7(d)].The polaronlike structure was reproduced with differentlattice spacing, and the effect became more pronouncedwhen the lattice spacing was reduced to �40 �m (Fig. 7,bottom panel). The observed process is quite similar to

that caused by a polaron in solid-state physics in whichan electron drags and dislocates heavy ions as it propa-gates through an ionic crystal.38 Since our lattice itself ispartially coherent and its spacing is not smaller than thecoherence length, nearby pixels in the lattice have no, oronly weak, initial phase correlation. In addition the lat-tice is mutually incoherent with the probe. Thus onewould expect only attraction between nearby lattice sitesrather than repulsion from incoherent soliton interac-tions. Furthermore, since the closest four neighboringsolitons are initially equally spaced around the centralone at which the probe beam was aimed, the observed be-havior cannot be attributed simply to the anomalous in-teraction between photorefractive solitons in which at-traction or repulsion depends merely on soliton mutualseparation.39 Instead the observed polaronlike structuresuggests that the probe beam might have induced a cer-tain degree of coherence in the neighboring lattice siteswith different phase correlation through interaction. Thisissue certainly merits further investigation.

4. OPTICAL VORTICES IN LINEAR ANDNONLINEAR PHOTONIC LATTICESA. Formation of Discrete Vortex Solitons in LinearLatticesAn important nonlinear phenomenon in 2D lattices isthe propagation of optical beams with complex internalstructure on the lattice, e.g., the propagation of opticalvortices carrying orbital angular momentum. Here wepresent our experimental results of a vortex beam asprobe propagating in a 2D linear waveguide lattice. Inthis case, the lattice beam is o polarized while the vortexbeam is e polarized. Typical experimental results are

Fig. 7. Formation of soliton-induced polaronlike structures as the intensity of the control soliton was increased gradually. The intensityratio between the probe soliton and the lattice is (a) 0.1, (b) 0.6, (c) 1.0, (d) 1.5. The lattice spacing is about 55 �m for the top panels and40 �m for the bottom panels. The images in the bottom panels have been enlarged for better visualization of the central part of thelattice.




PROO

F COPY [57713] 014507JO

B shown in Fig. 8, for which a waveguide lattice with 28 �mspacing is created. The vortex beam with unit topologicalcharge [normal diffraction shown in Fig. 8(a)] is thenlaunched straight into the middle of the four lattice sites,so that the vortex center sits right in the middle of fourlattice sites (off-site excitation20,21). As a result of wave-guide coupling, the vortex beam undergoes discrete dif-fraction when the nonlinearity is low [Fig. 8(b) and 8(c)],but it evolves into a discrete vortex soliton at an appro-priate level of high nonlinearity [Fig. 8(d)] with most ofthe energy concentrated at the central four sites along theprincipal axes of the lattice. To confirm the nontrivialphase of the vortex soliton, a weak reference beam is in-troduced to interfere with the discrete vortex after it exitsthe crystal. We use a piezotransducer mirror in the refer-ence beam path to control its phase relative to the vortexbeam. As we move the piezotransducer mirror, a series ofinterferograms is recorded to reconstruct the phase of thevortex. Examples of the interferograms are presented inFigs. 8(f)–8(h), which show that one of the four lobes in-creases its intensity, whereas the corresponding diagonallobe decreases its intensity. Furthermore, the lobe withthe strongest intensity is alternating among the fourspots. These interferograms confirm that the four lobes ofthe output vortex have a nontrivial phase relation—a � /2phase ramp—as expected for the discrete vortex soliton.By removing the reference beam, the discrete vortex soli-ton is restored in steady state [Fig. 8(e)].

B. Interaction of a Vortex with a Nonlinear LatticeFinally, we present our experimental results on interac-tion of a vortex beam with a nonlinear lattice induced by2D, pixel-like spatial solitons. In this case, both the probeand the lattice beams are e polarized. When the lattice isoperated in the nonlinear regime, it can acquire linear orangular momentum when interacting with other beams.The deformation of the lattice caused by soliton-lattice in-teraction as shown in Figs. 4–6 can be considered ex-amples of transferring linear transverse momentum from

a Gaussian beam to the lattice. Here we address the issueof interaction between a solitonic lattice and beams car-rying angular momentum. This is a nontrivial issue, as inperiodic structures or lattices the continuous rotationalsymmetry of homogeneous media is broken, thus angularmomentum is not generally conserved. In the case of adiscrete vortex-ring soliton presented in Fig. 8, it is thecombination of “discrete” dynamics and nonlinearity thatallows the soliton to maintain its vortex phase.20,21,40,41

However, higher-order vortices may not conserve their to-pological charges even in the linear lattice.42 It is thusnatural to ask: Can the angular momentum carried by avortex beam be transferred to a nonlinear lattice?

To answer this question, we launched a vortex beam ofopposite topological charges into a nonlinear lattice. Theexperimental results are summarized in Fig. 9, for whicha solitonic lattice of about 40 �m spacing was first cre-ated. The input size of the vortex ring was adjusted byproper imaging to cover the central four lattice sites [seeFig. 9(a)]. After linear propagation through the 8 mm-longcrystal the vortex ring expanded to about 2.5 times in itsdiameter, while the propagation direction of the vortexcore was unchanged (collinear with the lattice propaga-tion). After a bias field of 2.2 kV/cm was applied to thecrystal, the vortex beam broke up into two major fila-ments as a result of azimuthal instability, each of whichrotated slightly in direction corresponding to the sign oftopological charge. Such breakup can be seen clearly inFig. 9(b), where the two filaments rotated counterclock-wise and clockwise for vortices of opposite topologicalcharge m= +1 and m=−1, respectively. The simultaneouspropagation of the lattice and the vortex, however, dras-tically influenced the vortex dynamics. Because of themodulation–interaction offered by the nonlinear lattice,the vortex ring broke up into four filaments [Fig. 9(c)]rather than two, similar to the vortex propagation in alinear lattice [Fig. 8]. In contradistinction to that in a lin-ear lattice, the four spots resulting from the vortexbreakup rotated from their initial orientation, which also

Fig. 8. Vortex beam propagating in an optically-induced linear lattice showing formation of a 2D discrete vortex soliton. Top panelsshow the 3D intensity plots of the vortex beam at crystal output of (a) normal diffraction without the lattice (b), (c) discrete diffractionat low nonlinearity, (d) discrete soliton at high nonlinearity (bias field, 3000 V/cm). The lattice spacing is 28 �m. Bottom panels show (e)discrete vortex soliton reproduced in a lattice of 20 �m spacing, (f)–(h) a series of interferograms between the vortex soliton and a planewave confirming the nontrivial spiral phase structure of the vortex.


APC:#3



PROO

F COPY [57713] 014507JO

B

caused lattice twisting in the area of interaction. As seenin Fig. 9(d) the direction of twisting is opposite for vorti-ces with opposite topological charges. Through these ex-periments we believe that the lattice acquired some angu-lar momentum from the vortex beam, which was thenredistributed to all the lattice sites. This behavior wasmore clearly visible in the transient dynamical evolutionby taking advantage of the slow response of the photore-fractive nonlinearity. Right after launching the vortexbeam into the lattice, we observed that the correspondinglattice sites were strongly twisted, but the twisting waslater reduced through interaction with the neighboringlattice sites.

While it is possible to confine the high-order vortices ina linear lattice to form localized states ofquasi-vortices,42,43 the nonlinear interaction between ahigh-order vortex and a solitonic lattice often leads to dis-ordered structures. Experiments were performed with acharge-2 and a charge-4 vortex in a manner similar tothat with the charge-1 vortex. Vortex-induced twisting ofthe interacting lattice sites was observed, but the high-order vortices broke up into more filaments that tended to

escape from the four lattice sites at which the vortex wasinitially aimed. Figure 10 shows the preliminary resultsobtained with a charge-2 vortex. Although vortex-inducedlattice twisting was clearly observed in a transient state,the outcome of vortex-lattice nonlinear interaction insteady state is very sensitive to the initial symmetry ofthe vortex ring, the uniformity of the lattice, and thealignment of the vortex–lattice beams.

We note that angular-momentum-carrying beams oroptical vortices are known to be useful for manipulation ofmicroparticles.44,45 Since a nonlinear lattice can be con-sidered a lattice of solitonic particles, it is reasonable toexpect that the orbital angular momentum can be corre-spondingly transferred from a vortex beam to a solitoniclattice. However, solitons as formed in a nonlinear latticeare not free to move, and the lattice typically breaks therotational symmetry of homogeneous media. Thus angu-lar momentum is not generally conserved in the process ofvortex–lattice interaction. We believe that the interactioncould lead to vortex-induced defect states in a nonlinearlattice, and it might also be possible to form a mutuallytrapped bound state between the high-order vortices and

Fig. 9. Vortex beam of unit topological charge interacting with an optically-induced nonlinear lattice showing lattice twisting driven bythe angular momentum carried by the vortex beam. (a) Input intensity patterns of the lattice (top) and the vortex (bottom). Nonlinearoutput of the vortex propagating (b) alone, (c) with the lattice. (d) Nonlinear output of the lattice corresponding to (c) when the vortex hasa positive (top) or a negative (bottom) charge. The lattice spacing is 40 �m in all cases.

Fig. 10. Doubly-charged vortex beam �m= +2� interacting with a nonlinear solitonic lattice. (a) Input intensity pattern of the vortex.Nonlinear output of the vortex propagating (b) alone, (c) with the lattice. (d) Nonlinear output of the lattice propagating with the vortex;the corresponding unperturbed lattice is shown in Fig. 2(c). The lattice spacing is 37 �m in all cases.




PROO

F COPY [57713] 014507JO

B the lattice. Further experimental and numerical studiesare currently under way to better understand the dynam-ics of vortex–lattice interaction.

5. SUMMARYWe have demonstrated experimentally a number of novelphenomena associated with soliton–lattice and vortex–lattice interactions in a 2D, optically-induced, nonlinearphotonic lattice, including lattice dislocation and creationof structures akin to optical polarons. We have observedsoliton-induced lattice deformation–compression whenthe interacting soliton has an initial linear transversemomentum, as well as vortex-induced lattice twisting dueto the transfer of angular momentum carried by the vor-tex. In addition 2D fundamental discrete solitons or dis-crete vortex solitons have been successfully demonstratedwhen a Gaussian beam or a vortex beam is launched ap-propriately into an optically induced linear lattice. Ourresults may pave the way for observation of similar phe-nomena in other types of nonlinear periodic systems, suchas Bose–Einstein condensates loaded onto an optical lat-tice.

ACKNOWLEDGMENTSZ. Chen is also with the Applied Physical School, NankaiUniversity, Tianjin, China. This work was supported byAFOSR, ARO , the Overseas Excellent Young ResearcherAward from NSF of China (No. 60328406), the PittsburghSupercomputing Center, and the Australian ResearchCouncil. We thank J. Yang, E. D. Eugenieva, M. Segev,P.G. Kevrekidis, T.J. Alexander, E. A. Ostrovskaya, I.Makasyuk, J. Young, W. Krolikowski, and J. Xu for assis-tance and numerous discussions.

Corresponding author Z. Chen’s e-mail address [email protected].

REFERENCES1. D. N. Christodoulides, F. Lederer, and Y. Siberberg,

“Discretizing light behaviour in linear and nonlinearwaveguide lattices,” Nature (London) 424, 817–XXX(2003).

2. D. Campbell, S. Flach, and Y. S. Kivshar, “Localizingenergy through nonlinearity and discreteness,” Phys. Today57, 43–XX (2004).

3. D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt.Lett. 13, 794–XXX (1988).

4. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd,and J. S. Aitchison, “Observation of discrete solitons inoptical waveguide arrays,” Phys. Rev. Lett. 81, 3383–XXX(1998).

5. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg,and Y. Silberberg, “Dynamics of discrete solitons in opticalwaveguide arrays,” Phys. Rev. Lett. 83, 2726–XXX (1999).

6. A. J. Sievers and S. Takeno, “Intrinsic localized modes inanharmonic crystals,” Phys. Rev. Lett. 61, 970–XXX (1988).

7. A. Trombettoni and A. Smerzi, “Discrete solitons andbreathers with dilute Bose–Einstein Condensates,” Phys.Rev. Lett. 86, 2353–XXXX (2001).

8. D. N. Christodoulides and E. D. Eugenieva, “Blocking androuting discrete solitons in two-dimensional networks ofnonlinear waveguide arrays,” Phys. Rev. Lett. 87,233901–XX? (2001).

9. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W.

Fleischer, and M. Segev, “Discrete solitons inphotorefractive optically induced photonic lattices,” Phys.Rev. E 66, 046602–XX? (2002).

10. J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, andD. N. Christodoulides, “Observation of discrete solitons inoptically induced real time waveguide arrays,” Phys. Rev.Lett. 90, 023902–XX? (2003).

11. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N.Christodoulides, “Observation of two-dimensional discretesolitons in optically induced nonlinear photonic lattices,”Nature (London) 422, 147–XXX (2003).

12. D. Neshev, E. Ostrovskaya, Y. Kivshar, and W.Krolikowski, “Spatial solitons in optically inducedgratings,” Opt. Lett. 28, 710–XXX (2003).

13. H. Martin, E. D. Eugenieva, Z. Chen, and D. N.Christodoulides, “Discrete solitons and soliton-induceddislocations in partially-coherent photonic lattices,” Phys.Rev. Lett. 92, 123902–XX? (2004).

14. J. Klinger, H. Martin, and Z. Chen, “Experiments oninduced modulational instability of an incoherent opticalbeam,” Opt. Lett. 26, 271–XXX (2001).

15. Z. Chen and K. MaCarthy , “Spatial soliton pixels frompartially coherent light,” Opt. Lett. 27, 2019–XXXX (2002).

16. J. Petter, J. Schrder, D. Trger, and C. Denz, “Opticalcontrol of arrays of photorefractive screening solitons,” Opt.Lett. 28, 438–XXX (2003).

17. M. Petrovic, D. Traqer, A. Strinic, M. Belic, J. Schoder, andC. Denz, “Solitonic lattices in photorefractive crystals,”Phys. Rev. E 68, 055601(R)–XXX (2003).

18. D. N. Neshev, Y. S. Kivshar, H. Martin, and Z. Chen,“Soliton stripes in two-dimensional nonlinear photoniclattices,” Opt. Lett. 29, 486–XXX (2004).

19. A. S. Desyatnikov, E. A. Ostrovskaya, Y. S. Kivshar, and C.Denz, “Composite band-gap solitons in nonlinear opticallyinduced lattices,” Phys. Rev. Lett. 91, 153902 (2003).

20. D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S.Kivshar, H. Martin, I. Makasyuk, and Z. Chen,“Observation of discrete vortex solitons in optically-inducedphotonic lattices,” Phys. Rev. Lett. 92, 123903 (2004).

21. J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev,J. Hudock, and D. N. Christodoulides, “Observation ofvortex-ring discrete solitons in 2D photonic lattices,” Phys.Rev. Lett. 92, 123904 (2004).

22. Z. Chen, H. Martin, E. D. Eugenieva, J. Xu, and A.Bezryadina, “Anisotropic enhancement of discretediffraction and formation of two-dimensional discrete-soliton trains,” Phys. Rev. Lett. 92, 143902 (2004).

23. Z. Chen, I. Makasyuk, A. Bezryadina, and J. Yang,“Observation of two-dimensional lattice vector solitons,”Opt. Lett. 29, 1656–XXX (2004).

24. J. Yang, I. Makasyuk, A. Bezryadina, and Z. Chen, “Dipolesolitons in optically-induced two-dimensional photoniclattices,” Opt. Lett. 29, 1662–XXXX (2004).

25. H. Buljan, O. Cohen, J. W. Fleischer, T. Schwartz, M.Segev, Z. H. Musslimani, N. K. Efremidis, and D. N.Christodoulides, “Random-phase solitons in nonlinearperiodic lattices,” Phys. Rev. Lett. 92, 223901 (2004).

26. J. Meier, G. I. Stegeman, D. N. Christodoulides, Y.Silberberg, R. Morandotii, H. Yang, G. Salamo, M. Sorel,and J. S. Aitchison, “Experimental observation of discretemodulational instability,” Phys. Rev. Lett. 92, 163902(2004).

27. D. Mandelik, R. Morandotti, J. S. Aitchison, and Y.Silberberg, “Gap solitons in waveguide arrays,” Phys. Rev.Lett. 92, 093904 (2004).

28. D. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski,and Y. S. Kivshar, “Controlled generation and steering ofspatial gap solitons,” Phys. Rev. Lett. 93, 083905 (2004).

29. O. Cohen, B. Freedman, J. W. Fleischer, M. Segev, and D.N. Christodoulides, “Grating-mediated waveguiding,” Phys.Rev. Lett. 93, 103902 (2004).

30. J. Meier, G. I. Stegeman, Y. Silberberg, R. Morandotii, andJ. S. Aitchison, “Nonlinear optical beam interactions inwaveguide arrays,” Phys. Rev. Lett. 93, 093903 (2004).

31. M. Mitchell, Z. Chen, M.-F. Shih, and M. Segev, “Self-


APC:#4

APC:#5

APC:#6



PROO

F COPY [57713] 014507JO

B trapping of partially spatially incoherent light”, Phys. Rev.Lett. 77, 490–(1996).

32. Z. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. N.Christodoulides, “Self-trapping of dark incoherent lightbeams”, Science 280, 889–XXX (1998).

33. M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A.Yariv, “Steady-state spatial screening solitons inphotorefractive materials with external applied field,”Phys. Rev. Lett. 73, 3211–XXXX (1994).

34. M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides,and A. Vishwanath, “Modulation instability of incoherentbeams in noninstantaneous nonlinear media,” Phys. Rev.Lett. 84, 467–XXX (2000).

35. T. H. Coskun, D. N. Christodoulides, Y.-R. Kim, Z. Chen, M.Soljacic, and M. Segev, “Bright spatial solitons on apartially incoherent background,” Phys. Rev. Lett. 84,2374–XXXX (2000).

36. D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N.Christodoulides, “Modulation instability and patternformation in spatially incoherent light beams,” Science290, 495–XXX (2000).

37. S. R. Singh and D. N. Christodoulides, “Evolution of spatialoptical solitons in biased photorefractive media understeady state conditions,” Opt. Commun. 118, 569–XXX(1995).

38. C. Kittel, Introduction to Solid-State Physics (Wiley, 1996).

39. W. Krolikowski, M. Saffman, B. Luther-Davies, and C.Denz, “Anomalous interaction of spatial solitons inphotorefractive media,” Phys. Rev. Lett. 80, 3240–XXXX(1998).

40. B. A. Malomed and P. G. Kevrekidis, “Discrete vortexsolitons,” Phys. Rev. E 64, 026601–XX? (2001).

41. J. Yang and Z. H. Musslimani, “Fundamental and vortexsolitons in a two-dimensional optical lattice”, Opt. Lett. 28,2094–XXXX (2003).

42. P. G. Kevrekidis, B. A. Malomed, Z. Chen, and D. J.Frantzeskakis, “Stable higher-order vortices and quasi-vortices in the discrete nonlinear Schrödinger equation,”Phys. Rev. E 70, 056612 (2004).

43. J. Yang, I. Makasyuk, P. G. Kevrekidis, H. Martin, B. A.Malomed, D. J. Frantzeskakis, and Z. Chen, “Necklace-likesolitons in optically induced photonic lattices,” Phys. Rev.Lett. XX, XXX (200X).

44. H. He, M. E. J. Friese, N. R. Heckenberg, and H.Rubinsztein-Dunlop, “Direct observation of transfer ofangular momentum to absorptive particles from a laserbeam with a phase singularity,” Phys. Rev. Lett. 75,826–XXX (1995).

45. M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R.Heckenberg, “Optical angular-momentum transfer totrapped absorbing particles,” Phys. Rev. A 54, 1593–XXX(1996).




PROO

F COPY [57713] 014507JO

B AUTHOR QUERIES — 014507JOB

#1 AQ: Can’t figure this out. Please explain orcorrect.

#2 AQ: As meant?#3 AQ: As meant?

#4 AQ: Please spell out AFOSR, ARO, and NSF.#5 AQ: Please provide page range for ALL refs.,

where applicable.#6 AQ: Spelling correct.

NOT FOR PRINT! FOR REVIEW BY AUTHOR NOT FOR PRINT!


Experiments on Gaussian beams and vortices in optically induced photonic...

Documents

Transcript of Experiments on Gaussian beams and vortices in optically induced photonic...